# Mathematics

*Major: **Mathematics*

*Degree Awarded: *Master of Science (MS) or Doctor of Philosophy (PhD)

*Calendar Type: Quarter*

*Total Credit Hours:* 45.0 (MS) or 90.0 (PhD)

*Classification of Instructional Programs (CIP) code:* 27.0101

*Standard Occupational Classification (SOC) code:** 15-2021; 15-2041*

## About the Program

The Department of Mathematics is a broadly based academic unit offering instructional programs and carrying on research activities in mathematics. Doctor of Philosophy and Master of Science degrees are offered.

Areas of research specialty among the faculty include applied mathematics, algebraic combinatorics, biomathematics, discrete mathematics, optics, analysis, number theory, numerical analysis, probability and statistics, matrix and operator theory, fluid mechanics, and partial differential equations.

#### Additional Information

For more information about theses graduate programs, visit Department of Mathematics webpage.

## Admission Requirements

Applicants should hold a BS degree in mathematics or the equivalent and meet the University's graduate admission standards. In particular, the student should have had intensive exposure to proof oriented courses, such as real analysis and abstract algebra. Students requesting financial aid are required to take the Graduate Record Examination General Test. Because many of the core courses are two- or three-term sequences beginning in the fall, new students are typically admitted to the programs only in the fall term. Admissions standards for the MS and PhD programs are equivalent.

For additional information on how to apply, visit Drexel University's Graduate Admissions website.

## Master of Science in Mathematics

Students must complete a minimum of 45.0 graduate credits for the MS degree. Of these 15 courses, the following six are required:

Required Courses | ||

MATH 504 | Linear Algebra & Matrix Analysis | 3.0 |

MATH 505 | Principles of Analysis I | 3.0 |

MATH 506 | Principles of Analysis II | 3.0 |

MATH 533 | Abstract Algebra I | 3.0 |

MATH 630 | Complex Variables I | 3.0 |

MATH 633 | Real Variables I | 3.0 |

The remaining 9 courses may be any graduate mathematics courses. In some cases, course substitutions may be made with courses from other departments. Elective courses taken outside the department must receive prior departmental approval in order to be counted toward the degree.

There are no thesis, language, or special examination requirements for the master's degree.

Students seeking a dual MS must satisfy core requirements for both degree programs.

Students should note that some departmental courses, such as Advanced Engineering Mathematics, are foundation courses and do not contribute to the departmental requirements for the degree. They do count toward the University requirements for a degree.

## PhD in Mathematics

Students must complete a minimum of 45 graduate credits for the PhD degree, in addition to the 45.0 required by the MS program for a total of 90.0 credits. Of the 45.0 credits of MS program courses, the following six are required:

Required Courses | ||

MATH 504 | Linear Algebra & Matrix Analysis | 3.0 |

MATH 505 | Principles of Analysis I | 3.0 |

MATH 506 | Principles of Analysis II | 3.0 |

MATH 533 | Abstract Algebra I | 3.0 |

MATH 630 | Complex Variables I | 3.0 |

MATH 633 | Real Variables I | 3.0 |

The remaining 27.0 credits, comprising the MS segment of the PhD program, may be any graduate mathematics courses. In some cases, course substitutions may be made with courses from other departments. Elective courses taken outside the department must receive prior departmental approval in order to be counted toward the degree.

The student must pass a written qualifying exam. The student is allowed two attempts. Students must take exam at the end of their first year, and have a second opportunity in September of their second year.

Students must take a PhD candidacy exam at the end of their second year. Additional coursework to reach the 90.0 credits required for the PhD will be agreed upon with the student's Graduate Advisor. Students should note that some departmental courses, such as MATH 544 Advanced Engineering Mathematics, are foundation courses and do not contribute to the departmental requirements for the degree. They do count toward the University requirements for a degree.

## Facilities

Department computers are accessible from residence halls over the campus network, and from off-campus via modem or an Internet Service Provider (ISP). Departmental and university networks provide access to the Internet and the Pennsylvania Education Network (PrepNET). Departmental research computers have a connection to the campus backbone at 100 Mbps and are also on the vBNS via a campus OCS ATM connection.

The computing resources of the Mathematics Department include:

- Math Resource Center (Korman 247): 6 Dell Optiplex (Core 2 Duo 2.8 Ghz, 3 GB RAM) running Windows XP Professional SP3.
- Faculty Center (Korman 207): 2 Lenovo ThinkCentre (Pentium 4 3.0 Ghz, 1 GB RAM) running Windows XP Professional SP3.
- Computer Server: One Penguin Server (Dual 2.2. GHz Opteron, 8 GB RAM) running Ubuntu Linux.
- File/Print/Mail/Web Server: 2 Penguin Servers (Dual 2.8 GHz Zeon, 1 GB RAM) running Ubuntu Linux and connected to 600GB RAID 5 Disk over a fully switched gigabit Ethernet network, 2TB mirrored RAID.

### Courses

**MATH 504 Linear Algebra & Matrix Analysis 3.0 Credits**

Course topics include the QR decomposition, Schur's triangularization theorem, the spectral decomposition for normal matrices, the Jordan canonical form, the Courant-Fisher theorem, singular value and polar decompositions, the Gersgorin disc theorem, the Perron-Frobenius theorem, and other current matrix analysis topics. Applications of the material are outlined as well.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 505 Principles of Analysis I 3.0 Credits**

Metric spaces, compactness, connectedness, completeness. Set theory and cardinality, continuity, differentiation, Riemann integral.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 506 Principles of Analysis II 3.0 Credits**

A continuation of MATH 505. Uniform convergence, Fourier series, Lebesque integral in Euclidean spaces, differential calculus in Euclidean spaces, inverse and implicit functions theorems, change of variables in multiple integrals.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 505 [Min Grade: C]

**MATH 507 Applied Mathematics I 3.0 Credits**

Covers matrix theory, linear transformations, canonical forms, matrix decompositions, and factorizations, including the singular value decomposition, quadratic forms, matrix least squares problems, and fast unitary transforms. Introduces computational linear algebra.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 508 Applied Mathematics II 3.0 Credits**

Covers the techniques of mathematical modeling in the physical and biological sciences using discrete and combinatorial mathematics, probabilistic methods, variational principles, Fourier series and integrals, integral equations, calculus of variations, asymptotic series and expansions, and eigenvalue problems associated with Sturm-Liouville boundary value problems. Topics vary from year to year.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 507 [Min Grade: C]

**MATH 509 Applied Mathematics III 3.0 Credits**

Continues the theme of MATH 508. Topics vary from year to year.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 508 [Min Grade: C]

**MATH 510 Applied Probability and Statistics I 3.0 Credits**

Covers basic concepts in applied probability; random variables, distribution functions, expectations, and moment generating functions; specific continuous and discrete distributions and their properties; joint and conditional distributions; discrete time Markov chains; distributions of functions of random variables; probability integral transform; and central limit theorem.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 511 Applied Probability and Statistics II 3.0 Credits**

Covers probability plots and graphical techniques for determining distribution of data, including sampling and sampling distributions, law of large numbers, parametric point estimation, maximum likelihood estimation, Bayes estimation, properties of estimators, sufficient statistics, minimum variance unbiased estimators, and parametric interval estimation. Introduces hypothesis testing.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 510 [Min Grade: C]

**MATH 512 Applied Probability and Statistics III 3.0 Credits**

Covers hypothesis testing, analysis of variance, multiple regression, and special topics. Introduces linear models.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 511 [Min Grade: C]

**MATH 520 Numerical Analysis I 3.0 Credits**

Covers polynomial interpolation, numerical solutions of nonlinear equations, numerical integration (Newton-Cotes, Gauss quadrature), error estimates of various numerical methods, and function approximation (polynomial, Fourier, Pade).

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 521 Numerical Analysis II 3.0 Credits**

Covers numerical linear algebra and matrix computation, direct and iterative methods for solving linear systems and eigenvalue problems, least square problems, various matrix factorizations (QR, singular value decomposition, LU and Cholesky), and Krylov subspace methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 522 Numerical Analysis III 3.0 Credits**

Covers numerical solutions of ordinary and partial differential equations.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 520 [Min Grade: C]

**MATH 523 Computer Simulation I 3.0 Credits**

Covers computer simulation of pseudo-random variables, including Monte Carlo methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 510 [Min Grade: C]

**MATH 524 Computer Simulation II 3.0 Credits**

Covers discrete and continuous event simulation models and techniques.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 523 [Min Grade: C]

**MATH 525 Topics in Computer Simulation 3.0 Credits**

Covers statistical analysis of simulation data, variance reduction techniques, and advanced topics in simulation.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 524 [Min Grade: C]

**MATH 530 Combinatorial Mathematics I 3.0 Credits**

Covers graphs and networks, with an emphasis on algorithms. Includes minimum spanning trees, shortest path problems, connectivity, network flows, matching theory, Eulerian and Hamiltonian tours, graph coloring, and random graphs.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 531 Combinatorial Mathematics II 3.0 Credits**

Covers mathematical tools for the analysis of algorithms, including combinatorics, recurrence relations and generating functions, elementary asymptotics, and probabilistic methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 530 [Min Grade: C]

**MATH 532 Topics in Combinatorial Math 3.0 Credits**

Covers topics in discrete mathematics, including asymptotic enumeration, number theory, probabilistic combinatorics, and combinatoric algorithms.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 531 [Min Grade: C]

**MATH 533 Abstract Algebra I 3.0 Credits**

Covers groups, transformation groups and group actions, isomorphism and homomorphism theorems, Sylow theorems, symmetric groups, rings, and fields.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 534 Abstract Algebra II 3.0 Credits**

Covers factorization domains, Euclidean domains, and polynomial rings, and modules, vector spaces, and linear transformations.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 533 [Min Grade: C]

**MATH 535 Topics in Abstract Algebra 3.0 Credits**

This third course in the Abstract Algebra sequence covers a selection of topics in advanced modern algebra such as symmetries, representation theory, algebraic geometry, homological algebra, Galois Theory and coding theory.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated 3 times for 9 credits

**Prerequisites:**MATH 533 [Min Grade: C] and MATH 534 [Min Grade: C]

**MATH 536 Topology I 3.0 Credits**

Covers general topological spaces, metric spaces, and function spaces; open sets, limit points, limits of sequences, convergence, separation axioms, compactness, connectedness, continuity, homeomorphisms, and product of N-spaces; and specialized applications to the real line, Euclidean N-space, and well-known function spaces.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 537 Topology II 3.0 Credits**

Continues MATH 536.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 538 Manifolds 3.0 Credits**

Topics will be selected from the following: Differential structures, immersion theorems, tangent bundles, vector fields and distributions, integral manifolds, integration on manifolds, differential forms, general Stokes Theorem, applications to physics and engineering.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 540 Numerical Computing 3.0 Credits**

Intended to introduce students to contemporary computing environments and the associated tools. Uses contemporary software tools and specific applications from science and engineering to illustrate numerical and visualization methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 544 Advanced Engineering Mathematics I 3.0 Credits**

Covers solution techniques for ordinary differential equations, including series techniques, Legendre and Bessel functions, Sturm-Liouville theory, and Laplace and Fourier techniques. Introduces symbolic computation as time permits.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 545 Advanced Engineering Mathematics II 3.0 Credits**

Covers partial differential equations, including separation of variables and its applications to standard equations. Introduces Green's functions for differential equations.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 544 [Min Grade: C]

**MATH 546 Advanced Engineering Mathematics III 3.0 Credits**

Covers complex analysis, including complex differentiation and integration, Cauchy's theorems and residue theory, and their applications; conformal maps; and applications to fluid flow.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 545 [Min Grade: C]

**MATH 553 Sci Comp & Visualization I 3.0 Credits**

Covers scientific computing, with an emphasis on numerical computing and visualization techniques. Includes techniques of computational geometry, including an introduction to methods used to describe the shapes of free-form curves, surfaces, and volumes, and applications to computer-aided design and other areas of scientific computing.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 540 [Min Grade: C]

**MATH 554 Sci Comp & Visualization II 3.0 Credits**

Covers scientific visualization, using a computational environment that includes high-performance workstations and supercomputers, and application in science and engineering. Includes applications to finite element and difference methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 553 [Min Grade: C]

**MATH 555 Topics in Sci Comp & Visualiz 3.0 Credits**

Covers special topics chosen from contemporary problem areas in scientific computing and visualization, including digital image processing, wavelet transforms and their numerical treatment, numerical conformal mapping, and contemporary problem areas in scientific computing and visualization.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 554 [Min Grade: C]

**MATH 572 Financial Mathematics: Fixed Income Securities 3.0 Credits**

The course is a mathematical introduction to interest rates and interest rates related instruments including loans, bonds, mortgages and swaps. The course emphasizes the mathematical aspects of the subject and computational implementation.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 610 Probability Theory I 3.0 Credits**

Covers basics of modern probability theory: properties of probability measures, independence, Borel-Cantelli lemma, zero-one law, random variables, distribution theory, and expectations.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 633 [Min Grade: C]

**MATH 611 Probability Theory II 3.0 Credits**

Covers further development of modern probability theory, including modes of convergence of random variables, series of random variables, weak and strong laws of large numbers, characteristics functions, inversion formula and continuity theorem, and central limit theorem.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 610 [Min Grade: C]

**MATH 612 Topics in Probability Theory 3.0 Credits**

This third course in the probability sequence covers a selection of topics in modern probability theory. Topics may include: theory of sums of independent random variables, inequalities, martingale theory, combinatorial probability.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated 2 times for 6 credits

**Prerequisites:**MATH 611 [Min Grade: C]

**MATH 613 Stochastic Processes I 3.0 Credits**

Covers conditional probabilities, expectations, Markov chains, classification of states, recurrence and absorption probabilities, asymptotic behavior, random walk, birth and death processes, and ruin problems.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 510 [Min Grade: C] and MATH 611 [Min Grade: C]

**MATH 614 Stochastic Processes II 3.0 Credits**

Covers queuing theory, waiting line models, embedded Markov chain method, and optimization problems. Includes applications and simulation.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 613 [Min Grade: C]

**MATH 615 Topics in Stochastic Processes 3.0 Credits**

Covers topics including branching processes, Brownian motion, renewal processes, compounding stochastic processes, martingales, and decision-making under uncertainty.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 613 [Min Grade: C]

**MATH 620 Partial Differential Equations I 3.0 Credits**

Covers derivation and classification of partial differential equations; elementary methods of solution, including Fourier series and transform techniques; linear and equilinear equations of the first order; hyperbolic, elliptic, and parabolic type equations; maximum principles; existence, uniqueness, and continuous dependence theorems; Riemann's method; method of characteristics; Green's functions; and variational and numerical methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 621 Partial Differential Equations II 3.0 Credits**

Continues MATH 620.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 622 Partial Differential Equations III 3.0 Credits**

Continues MATH 621.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 623 Ordinary Differential Equations I 3.0 Credits**

Covers existence and uniqueness theorems, properties of solutions, adjoint equations, canonical forms, asymptotic behavior, phase space, method of isocline, classification of singular points, linear two-dimensional autonomous systems, non-linear systems, stability theory, Lyapunov's methods, quadratic forms, construction of Lyapunov's function, boundedness, limit sets, applications to controls, linear equations with periodic coefficients, Floquet theory, characteristic multipliers and exponents, existence of periodic solutions to weakly non-linear systems, jump phenomena, subharmonic resonance, and stability of periodic solutions.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 624 Ordinary Differential Equations II 3.0 Credits**

Continues MATH 625.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 625 Ordinary Differential Equations III 3.0 Credits**

Continues MATH 626.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 630 Complex Variables I 3.0 Credits**

Covers Cauchy's theorem, Morera's theorem, infinite series, Taylor and Laurent explanations, residues, conformal mapping and applications, analytic continuation, and Riemann mapping theorem.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 631 Complex Variables II 3.0 Credits**

Covers entire functions, Picard's theorem, series and product developments, Riemann Zeta function, and elliptic functions.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 630 [Min Grade: C]

**MATH 632 Topics in Complex Variables 3.0 Credits**

Covers topics including global analytic functions, algebraic functions, and linear differential equations in the complex plane.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 631 [Min Grade: C]

**MATH 633 Real Variables I 3.0 Credits**

Covers algebra of sets, topology of metric spaces, compactness, completeness, function spaces, general theory of measure, measurable functions, integration, convergence theorems, and applications to classical analysis and integration.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 634 Real Variables II 3.0 Credits**

Covers Fubini's theorem, Radon-Nikodym theorem, LP-spaces, linear functionals on LP-spaces, Riesz-representation theorem, topological integration, Riesz-Markoz theorem, Luzin's theorem, basic complex functions, analytic functions, and complex-integration.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 633 [Min Grade: C]

**MATH 635 Real Variables III 3.0 Credits**

Covers topics including differentiation theory, Fourier series and transforms, and singular integrals.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 634 [Min Grade: C]

**MATH 640 Functional Analysis 3.0 Credits**

An introduction to abstract linear spaces, including normed linear spaces, Hilbert spaces, Banach spaces, and their duals. Fundamental theorems such as the Hahn-Banach theorem, open mapping and closed graph theorems will be covered, along with possible applications to differential and integral equations and fundamentals of distribution theory.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 504 [Min Grade: C] and MATH 506 [Min Grade: C]

**MATH 641 Harmonic Analysis 3.0 Credits**

Covers modern techniques and applications of harmonic analysis, including Fourier series, Fourier transforms and related topics.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 640 [Min Grade: C]

**MATH 642 Operator Theory 3.0 Credits**

An introduction to basic spectral theory of linear operators, theory of compact operators, and theory of unbounded operators.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 640 [Min Grade: C]

**MATH 643 Integral Equations I 3.0 Credits**

Covers theory and application of linear integral equations, including the Hilbert-Schmidt theory. Introduces non-linear and singular integral equations and numerical methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 645 Transform Theory I 3.0 Credits**

Covers selected topics from wavelet transforms, including properties; asymptotic analyses; and applications of the integral transforms of Laplace, Fourier, Mellin, and Radon.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 640 [Min Grade: C]

**MATH 646 Transform Theory II 3.0 Credits**

Covers selected topics from wavelet transforms and applications, convolution equations, and the calculus of distributions.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 640 [Min Grade: C] and MATH 645 [Min Grade: C]

**MATH 660 Lie Groups and Lie Algebras I 3.0 Credits**

Covers matrix groups, topological groups, locally isomorphic groups, universal covering groups, analytic manifolds, Lie groups; the Lie algebra of a Lie group, differential forms, and Lie's three theorems; analytic subgroups of a Lie group and compact Lie groups; and semisimple Lie algebras, general structure of Lie algebras, Cartan subalgebras, modules and representation, and computational techniques in representation theory.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 661 Lie Groups and Lie Algebras II 3.0 Credits**

Continues MATH 660.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 662 Lie Groups/Algebras III 3.0 Credits**

Continues MATH 661.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 670 Methods of Optimization I 3.0 Credits**

Provides a rigorous treatment of theory and computational techniques in linear programming and its extensions, including formulation, duality theory, simplex and dual-simplex methods, and sensitivity analysis; network flow problems and algorithms; systems of inequalities, including exploiting special structure in the simplex method and use of matrix decompositions; and applications in game theory and integer programming.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 671 Methods of Optimization II 3.0 Credits**

Covers necessary and sufficient conditions for unconstrained and constrained optimization. Includes computational methods including quasi-Newtonian and successive quadratic programming, and penalty and interior methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 672 Methods of Optimization III 3.0 Credits**

Covers advanced topics in mathematical programming, including interior point methods in linear programming; stochastic optimization; multi-objective optimization; and global minimax, functional, and non-linear least squares optimization methods.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 670 [Min Grade: C] and MATH 671 [Min Grade: C]

**MATH 673 Calculus of Variations 3.0 Credits**

Introduction to calculus of variations. Covers applications to geometry, classical mechanics and control theory, Euler-Lagrange equations, problems with constraints, canonical equations, Hamiltonian mechanics, symmetries and Noether's theorem, Hamilton-Jacobi theory, introduction to optimal control, maximum principle, and Hamilton-Jacobi-Bellman equations.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 701 Algebraic Combinatorics 3.0 Credits**

This course covers methods of Abstract Algebra that can be applied to various combinatorial problems and conversely, combinatorial methods to approach problems in representation theory, algebraic geometry, and homological algebra.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**Prerequisites:**MATH 533 [Min Grade: C] and MATH 534 [Min Grade: C]

**MATH 723 Mathematical Neuroscience 3.0 Credits**

This is an introduction to mathematical and computational techniques for analyzing neuronal models. Topics include conductance based models, neuronal excitability, bursting, neural networks, and compartmental models, as well as phase plane analysis, slow-fast systems, elements of applied bifurcation theory, and simulating differential equation models using MATLAB.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 898 Master's Thesis 0.5-20.0 Credits**

Master's thesis.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Not repeatable for credit

**MATH 997 Research 1.0-12.0 Credit**

Research.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH 998 Ph.D. Dissertation 1.0-12.0 Credit**

Ph.D. dissertation.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH I599 Independent Study in MATH 12.0 Credits**

Self-directed within the area of study requiring intermittent consultation with a designated instructor.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH I699 Independent Study in MATH 0.5-6.0 Credits**

Self-directed within the area of study requiring intermittent consultation with a designated instructor.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH I799 Independent Study in MATH 6.0 Credits**

Self-directed within the area of study requiring intermittent consultation with a designated instructor.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH I899 Independent Study in MATH 12.0 Credits**

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH I999 Independent Study in MATH 12.0 Credits**

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH T580 Special Topics in Mathematics 9.0 Credits**

Topics decided upon by faculty will vary within the area of study.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH T680 Special Topics in Mathematics 9.0 Credits**

Covers special topics of interest to students and faculty.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH T780 Special Topics in Mathematics 9.0 Credits**

Topics decided upon by faculty will vary within the area of study.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH T880 Special Topics in Mathematics 9.0 Credits**

Topics decided upon by faculty will vary within the area of study.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

**MATH T990 Special Topics in Mathematics 9.0 Credits**

Topics decided upon by faculty will vary within the area of study.

**College/Department:**College of Arts and Sciences

**Repeat Status:**Can be repeated multiple times for credit

## Mathematics Faculty

*(Duke University)*

*Associate Department Head, Mathematics*. Associate Professor. Applied analysis and computing for systems of nonlinear partial differential equations, especially free-surface problems in fluid dynamics.

*(Drexel University)*. Assistant Teaching Professor.

*(University of California at Berkeley)*. Associate Professor. Algebraic combinatorics, representation theory, and complexity theory.

*(University of Pennsylvania)*

*Associate Head of the Mathematics Department*. Professor. Functional analysis, C*-algebras and the theory of group representations.

*(University of Miami)*. Assistant Professor. Homological mirror symmetry, Landau-Ginzburg models, algebraic geometry, symplectic geometry.

*(Drexel University)*. Assistant Teaching Professor. Discrete mathematics and automata theory.

*(Drexel University)*. Assistant Teaching Professor.

*( University of Illinois at Chicago)*. Assistant Teaching Professor. Computational Fluid Dynamics, Free Boundary Problems.

*(Massachusetts Institute of Technology)*. Associate Professor. Intersection of physics, engineering, applied mathematics and computational science.

*(University of California at Berkeley)*. Assistant Teaching Professor. Function theory and operator theory, harmonic analysis, matrix theory.

*(University of Pittsburgh)*. Associate Professor. Biomathematics, dynamical systems, ordinary and partial differential equations and math education.

*(University of Pennsylvania)*. Professor. Geometry; optics; computer vision.

*(Warsaw University)*. Professor. Probability theory and its applications to analysis, combinatorics, wavelets, and the analysis of algorithms.

*(Drexel University)*. Assistant Teaching Professor.

*(The University of North Carolina at Chapel Hill)*. Visiting Assistant Professor. Algebraic Combinatorics and Algebraic Geometry--specifically positivity results for generating polynomials.

*(Kharkov University)*. Associate Professor. Operator theory, systems theory, complex analysis, C*-algebras and harmonic analysis.

*(University of Utah)*. Assistant Teaching Professor. Electromagnetic wave propagation in composite media, optimization and inverse problem.

*(York University)*. Assistant Teaching Professor. Algebraic combinatorics.

*(Boston University)*. Associate Professor. Ordinary and partial differential equations, mathematical neuroscience.

*(University of Houston)*. Visiting Assistant Professor. Inverse Problems

*(University of California, San Diego)*

*Undergraduate Advisor*. Professor. Algebraic combinatorics.

*(Rutgers University)*

*Department Head*. Professor. Partial differential equations and numerical analysis, including homogenization theory, numerical methods for problems with rough coefficients, and inverse problems.

*(Drexel University)*. Associate Teaching Professor.

*(Omsk State University)*. Associate Teaching Professor. Math education; geometrical modeling.

*(Drexel University)*. Instructor.

*(University of California at Berkeley)*. Associate Professor. Applied mathematics, numerical analysis, symbolic computation, differential geometry, mathematical physics.

*(University of California at Berkeley)*. Associate Professor. Applied mathematics, computed tomography, numerical analysis of function reconstruction, signal processing, combinatorics.

*(Drexel University)*. Associate Teaching Professor.

*(Drexel University)*. Teaching Professor. Probability and statistics.

*(University of Pennsylvania)*. Professor. Probabilistic combinatorics, asymptotic enumeration.

*(Rutgers University)*. Associate Professor. Discrete optimization, combinatorics, operations research, graph theory and its application in molecular biology, social sciences and communication networks, biostatistics.

*(Columbia University)*. Assistant Professor. Partial differential equations, scientific computing and applied mathematics.

*(Courant Institute, New York University)*. Professor. Homotopy theory, operad theory, quantum mechanics, quantum computing.

*(University of Kansas)*. Assistant Professor. Stochastic Calculus, Large Deviation Theory, Theoretical Statistics, Data Network Modeling and Numerical Analysis.

*(Boston University)*. Assistant Teaching Professor.

*(Harvard University)*. Assistant Teaching Professor. Applied statistics, data analysis, calculus, discrete mathematics, biostatistics.

*(Columbia University)*. Instructor.

*(Penn State University)*. Assistant Teaching Professor.

*(Vrije Universiteit, Amsterdam)*. Professor. Matrix and operator theory, systems theory, signal and image processing, and harmonic analysis.

*(Boston University)*

*Graduate Advisor*. Associate Professor. Partial differential equations, specifically nonlinear waves and their interactions.

*(Cornell University)*. Assistant Teaching Professor. Dynamical systems, neurodynamics.

*(Stanford University)*. Professor. Multiscale mathematics, wavelets, applied harmonic analysis, subdivision algorithms, nonlinear analysis, applied differential geometry and data analysis.

## Emeritus Faculty

*(University of Washington)*. Professor Emeritus. Functional analysis, wavelets, abstract harmonic analysis, the theory of group representations.

*(University of Pennsylvania)*. Professor Emeritus. Functional analysis, C*-algebras and group representations, computer science.

*(Temple University)*

*Dean Emeritus*. Professor Emeritus. Mathematics education, curriculum and instruction, minority engineering education.

*(Ohio State University)*. Associate Professor Emeritus. Number theory, approximation theory and special functions, combinatorics, asymptotic analysis.

*(University of Pennsylvania)*. Professor Emeritus. Lie algebras; theory, applications, and computational techniques; operations research.

*(University of California at Davis)*. Professor Emeritus. Probability and statistics, biostatistics, epidemiology, mathematical demography, data analysis, computer-intensive methods.

*(Courant Institute, New York University)*. Professor Emeritus. Applied mathematics, scattering theory, mathematical modeling in biological sciences, solar-collection systems.

*(University of Edinburgh)*. Professor Emeritus. Applied mathematics, special factors, approximation theory, numerical techniques, asymptotic analysis.

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